A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

开发了一种新方法来解决解决方案不平滑的最优控制问题。本文开发的方法采用了Legendre-Gauss-Radau正交直接配置方法的改进形式。与先前开发的标准Legendre-Gauss-Radau配置方法相比，此改良的Legendre-Gauss-Radau方法在网格间隔的末尾添加了两个变量和两个约束。另外两个变量是两个网格间隔之间的界面处的时间以及每个网格间隔结束时控件的时间。这两个附加约束是那些依赖于控件的微分方程的并置条件，以及在每个网格间隔的端点处的控件上的不等式约束。附加约束修改了非线性规划问题的搜索空间，从而获得了对非光滑度位置的精确近似。然后，开发了经过改进的Legendre–Gauss–Radau方法的变换后的伴随系统。利用这种变换的伴随系统，开发了一种将非线性规划问题的拉格朗日乘数变换为最优控制问题的代价的方法。此外，它表明，最昂贵的估计满足Weierstrass–Erdmann最优性条件之一。最后，以一个解决方案不光滑的例子来说明本文开发的方法。利用这种变换的伴随系统，开发了一种将非线性规划问题的拉格朗日乘数变换为最优控制问题的代价的方法。此外，它表明，最昂贵的估计满足Weierstrass–Erdmann最优性条件之一。最后，以一个解决方案不光滑的例子来说明本文开发的方法。利用这种变换的伴随系统，开发了一种将非线性规划问题的拉格朗日乘数变换为最优控制问题的代价的方法。此外，它表明，最昂贵的估计满足Weierstrass–Erdmann最优性条件之一。最后，以一个解决方案不光滑的例子来说明本文开发的方法。